New Process Dynamics and Control · 2020. 8. 27. · By: Gemechu Bushu (AAU) Process Dynamics and Control January 11, 2019 6 / 32 Modeling objectives is to describe process dynamics

Process Dynamics and Control

By: Gemechu Bushu (AAU)

January 11, 2019

By: Gemechu Bushu (AAU) Process Dynamics and Control January 11, 2019 1 / 32

Chapter Two

Theoretical Models of Chemical Processes

Mathematical Modeling is a mathematical abstraction of a real

process.

It is at best approximation of real process.

To analyze the behavior of a process, a mathematical representation

of the physical and chemical phenomenon taking place in it..

The activities leading to the construction of the model is called

modeling.


The main uses of mathematical modeling are:

To improve understanding of the process

To train plant operating personal

To design control strategy for new plant

To select controller settings

To design the controller law

To optimize process operating conditions


Six-step modeling procedure

Define goals

Prepare information

Formulate the model

Determine the solution

Analyze results

Validate the model


We apply this procedure

to many physical systems

overall material balance

component material balance

energy balances

Examples of variable selection

liquid level → total mass in liquid

pressure → total moles in vapor

temperature → energy balance

concentration → component mass


Overall Material Balance:

Accumulation of mass= Mass in - Mass out

Component Material Balance:

Accumulation of component mass= Component mass in - Component

mass out + Generation of component mass

State variables is a set of fundamental quantities whose value

describe the natural state of a given system.

State equations is a set of equations in the variables which describe

how the natural state of the given system change with time.


Modeling objectives is to describe process dynamics based on the laws

of conservation of mass, energy and momentum.

Mass Balance (Stirred tank)

Energy Balance (Stirred tank heater)

Momentum Balance (Car speed)

Degree of Freedom: Nf = Nv − Ne; where

NV is the total number of process variables, and

NE is the number of independent equations.


Mathematical Modeling of Common Chemical Processes

General balances taken in mathematical modeling

Total Mass Balance

dm

dt=

d(ρV )

dt=

∑i

ρiFi −∑j

ρjFj

Component Balance on A

dnAdt

=d(CAV )

dt=

∑i

CAiFi −∑j

CAjFj ± rAV

Total Energy Balance

dE

dt=

d(U + K + P)

dt=

∑i

ρiFihi −∑j

ρjFjhj ± Q


1) Mathematical Model: Surge tank (Liquid)

Modeling objective: Control of tank level

Assumptions: Incompressible flow



2) Mathematical Model: A Stirred Tank Heater

Modeling objective: Control of tank level and temperature

Assumptions: Incompressible flow, Tref = 0






3) Mathematical Model: A Stirred Tank Heater

Modeling objective: Control of tank level and temperature

Assumptions: Incompressible flow, Tref = 0

Perfect mixing; thus, the exit temperature T is also the temperature

of the tank contents.

The inlet and outlet flow rates are equal; thus, the liquid holdup V is

constant.

The density p and heat capacity C of the liquid are assumed to be

constant. 4. Heat losses are negligible.





The effluent flow rate F can be considered either as input or output.

If there is a control valve on the effluent stream so that its flow rate

can be manipulated by a controller, the variable

F is an input, since the opening of the valve is adjusted externally,

otherwise F is an output variable.By: Gemechu Bushu (AAU) Process Dynamics and Control January 11, 2019 20 / 32

4) Mathematical Model: A Stirred Tank Heater in jacketed Consider a

simple liquid phase, non-isothermal, irreversible, exothermic reaction:

A→ B where rA = kCαA , 4Hr = −λ kJ/kmol and the state variables are

V ,CA,T ,Tc .


The balance performed are

Total mass balance

Component balance on A

Energy balance inside the reactor

Energy balance around the reactor

The assumption taken are

First degree

reference temperature is zero

constant density

constant specific heat capacity

exothermic reaction






5) Mathematical Model: A Blending Process stirred-tank system

Develop the dynamics behavior of the process



Example 1: A stirred-tank blending process shown in figure below with a

constant liquid holdup is used to blend two streams whose densities (which

does not change during mixing) are both approximately 250kg/m3.

Assume that the process has been operating for a long period of time with

flow rates of ω1 = 300 kg/min and ω2 = 200 kg/min.

a) Develop the model that describes the dynamics behavior of the

process.

b) What is the steady-state value of output flow rate w?

c) If the output flow rate is 400kg/min, then what is the volume of the

tank at a period of 20 min?

d) If feed compositions (mass fractions) are x1 = 0.35 and x2 = 0.55,

then what is the steady-state value of composition, x?


Example 2: The liquid storage tank shown in figure below has two inlet

streams with mass flow rates ω1 and ω2 and an exit stream with flow rate

w. The cylindrical tank is 6.5m tall and 2m in diameter. The liquid has a

density of 800kg/m3. Normal operating procedure is to fill the tank until

the liquid level reaches a maximum value of the tank using constant flow

rates: ω1 = 250kg/min, ω2 = 150kg/min and ω = 300kg/min. Particular

day, corrosion of the tank has opened up a hole in the wall at a height of

2m, producing a leak whose volumetric flow rate q(m3/min) can be

approximated by: q = 0.0625√h − 2 where h is height in meters.

a) If the tank was initially empty, then determine the time to reach leak

point.

b) If mass flow rates ω1, ω2 and w are kept constant indefinitely, then

determine the maximum value of liquid level when no overflow occurs.By: Gemechu Bushu (AAU) Process Dynamics and Control January 11, 2019 30 / 32


Modeling Difficulties:

Poorly understood processes

Imprecisely known parameters

Size and complexity of a model


Documents

New Process Dynamics and Control · 2020. 8. 27. · By: Gemechu Bushu (AAU) Process Dynamics and Control January 11, 2019 6 / 32 Modeling objectives is to describe process dynamics